The Bass–Quillen conjecture: are all finitely generated projective modules over a polynomial ring with regular base extended from the base (the mixed-characteristic case)?

The Bass–Quillen conjecture predicts that for every commutative regular noetherian ring $A$ and every $n\geq 1$, each finitely generated projective module over the polynomial ring $A[t_1,\dots,t_n]$ is extended from $A$, i.e., is the base change of a finitely generated projective $A$-module. It is the natural extension of Serre's problem (solved by Quillen and Suslin when $A$ is a field) and is known for regular rings containing a field (Lindel, amplified by Popescu's general Néron desingularization), in dimension at most 2 (Murthy), and in dimension 3 with residue characteristic $\neq 2,3$ (Rao). The case of regular rings of mixed characteristic, especially ramified regular local rings with $p\in\mathfrak{m}^2$, remains open. Popescu (2018/2020) reduced the full conjecture to a single explicit family: it suffices to verify Bass–Quillen for the quotient rings $R/(p-b)$, where $R$ is a regular local ring essentially smooth over $\mathbb{Z}_{(p)}$ and $b\in\mathfrak{m}^2$.

Background

A module $P$ over a commutative ring $A$ is projective if it is a direct summand of a free module; projective modules are the algebraic counterpart of vector bundles, and classifying them is a central theme of algebraic K-theory. A noetherian ring is regular if every localization at a prime ideal is a regular local ring, i.e., a noetherian local ring $(R,\mathfrak{m})$ whose maximal ideal is generated by exactly $\dim R$ elements — the cleanest homological generalization of polynomial rings and smooth varieties. A module over $A[t_1,\dots,t_n]$ is extended (from $A$) if it is isomorphic to $A[t_1,\dots,t_n]\otimes_A Q$ for some $A$-module $Q$, meaning polynomial extension creates no genuinely new modules.

Serre's problem (1955) asked whether every finitely generated projective module over $k[t_1,\dots,t_n]$, with $k$ a field, is free; it was settled affirmatively and independently in 1976 by Quillen and by Suslin (Vaseršteĭn and Suslin gave a K-theoretic account the same year). In that work Quillen proved a patching theorem: a projective $A[t_1,\dots,t_n]$-module is extended iff its localizations at all maximal ideals of $A$ are extended, so the conjecture below reduces to regular local base rings. Bass (Problem IX, 1973) and Quillen then posed the natural generalization now called the Bass–Quillen (BQ) conjecture: for a regular ring $A$, every finitely generated projective $A[t_1,\dots,t_n]$-module is extended from $A$. Equivalently, passage to a polynomial ring leaves the category of vector bundles on $\operatorname{Spec} A$ unchanged (homotopy invariance for $K_0$).

The conjecture splits sharply by characteristic. A ring contains a field (equal characteristic) if some field lies inside it; otherwise it has mixed characteristic $(0,p)$, meaning $p$ is nonzero and noninvertible — the arithmetic situation, with $\mathbb{Z}_{(p)}$ (the integers localized at $p$) and $p$-adic rings as prototypes. For equal characteristic the conjecture holds: Lindel (1981) proved it for regular rings essentially of finite type over a field, and Popescu's general Néron desingularization theorem (every regular ring containing a field is a filtered colimit of smooth algebras) extends this to all regular rings containing a field. In mixed characteristic, a ring map is essentially smooth if it is a localization of a finitely presented smooth map; a mixed-characteristic regular local ring $(R,\mathfrak{m},k)$ with residue characteristic $p$ is unramified if $p\notin\mathfrak{m}^2$ and ramified otherwise. The unramified case was settled by work of Lindel and Swan for rings essentially of finite type over $\mathbb{Z}$, extended by Popescu. In dimension at most 2 the conjecture holds in full generality (Murthy 1966), and Rao (1988) proved the dimension-3 case when the residue characteristic is $\neq 2,3$.

The ramified mixed-characteristic case resisted direct attack. Swan observed in 1987 that it would help to know whether every regular local ring is a filtered inductive limit of regular local rings essentially of finite type over $\mathbb{Z}$ (limits along this filtration preserve the BQ property, since projective modules descend from some stage); Popescu proved a complete positive answer to Swan's question in 2018–2019, with an appendix by Česnavičius. However, this alone does not finish the conjecture: extending projective modules through the specific ring maps $A[W]/(W^e-b)$ appearing in the colimit construction is not automatic, and an example of Lam suggests an obstruction. Popescu therefore isolated (2018, published 2020) the remaining verification: if the BQ conjecture holds for every ring $R/(p-b)$ with $(R,\mathfrak{m})$ a regular local ring essentially smooth over $\mathbb{Z}_{(p)}$ and $b\in\mathfrak{m}^2$, then it holds for all regular rings. Adjacent lines — the non-archimedean analytic analogue of Kerz–Saito–Tamme, Euler class group theory (where BQ is used as a hypothesis), formal Laurent series reformulations, and the recent reductive-group torsor analogues of Guo–Liu and Stavrova — all treat the mixed-characteristic projective-module case as the outstanding open core. A detailed account of the equal-characteristic theory and history is in Lam's book on Serre's problem.

Problem Statement

The Bass–Quillen conjecture, in its natural generality: let $A$ be a commutative regular noetherian ring and $n\geq 1$. Is every finitely generated projective module $P$ over the polynomial ring $A[t_1,\dots,t_n]$ extended from $A$, i.e., is there a finitely generated projective $A$-module $Q$ with $P\cong A[t_1,\dots,t_n]\otimes_A Q$? By Quillen's patching theorem this is equivalent to the same statement for all regular local base rings, and the open part is mixed characteristic: regular rings not containing any field, where the equicharacteristic, unramified, dimension-$\leq 2$, and (residue characteristic $\neq 2,3$) dimension-3 cases are already known. An equivalent concentrated form of the same question (Popescu, Question 6 and Theorem 12 of arXiv:1810.00617): for every prime $p$, every regular local ring $(R,\mathfrak{m})$ essentially smooth over $\mathbb{Z}_{(p)}$, and every $b\in\mathfrak{m}^2$, is every finitely generated projective module over $R/(p-b)[T]$ extended from $R/(p-b)$? A positive answer for this family of rings is equivalent, by Popescu's theorem, to a positive answer for all regular rings. No restriction on dimension, residue characteristic, ramification, or answer form is imposed.

References

1. Daniel Quillen, 'Projective modules over polynomial rings', Inventiones Mathematicae 36 (1976), 167–171. DOI: 10.1007/BF01390008, https://doi.org/10.1007/BF01390008
2. L. N. Vaseršteĭn, A. A. Suslin, 'Serre's problem on projective modules over polynomial rings, and algebraic K-theory', Mathematics of the USSR-Izvestiya 10 (1976), 937–1001. DOI: 10.1070/IM1976v010n05abeh001822, https://doi.org/10.1070/im1976v010n05abeh001822
3. M. Pavaman Murthy, 'Projective A[X]-Modules', Journal of the London Mathematical Society s1-41 (1966), 453–456. DOI: 10.1112/jlms/s1-41.1.453, https://doi.org/10.1112/jlms/s1-41.1.453
4. Hartmut Lindel, 'On the Bass-Quillen conjecture concerning projective modules over polynomial rings', Inventiones Mathematicae 65 (1981), 319–323. DOI: 10.1007/BF01389017, https://doi.org/10.1007/bf01389017
5. Ravi A. Rao, 'Two examples of the Bass-Quillen-Suslin Conjectures', Mathematische Annalen 279 (1987), 227–238. DOI: 10.1007/BF01461720, https://doi.org/10.1007/bf01461720
6. Ravi A. Rao, 'The Bass-Quillen conjecture in dimension three but characteristic ≠ 2, 3 via a question of A. Suslin', Inventiones Mathematicae 93 (1988), 609–618. DOI: 10.1007/BF01410201, https://doi.org/10.1007/bf01410201
7. Dorin Popescu, 'On a question of Swan' (with an appendix by Kęstutis Česnavičius), Algebraic Geometry 6 (2019), no. 6, 716–729. DOI: 10.14231/AG-2019-030, arXiv:1803.06956, https://doi.org/10.14231/ag-2019-030
8. Dorin Popescu, 'On the Bass-Quillen Conjecture and Swan's question', arXiv:1810.00617 (2018); published as chapter in 'Combinatorial Structures in Algebra and Geometry', Springer Proceedings in Mathematics & Statistics, Springer (2020), 115–121. DOI: 10.1007/978-3-030-52111-0_9, https://doi.org/10.1007/978-3-030-52111-0_9
9. Moritz Kerz, Shuji Saito, Georg Tamme, 'Towards a non-archimedean analytic analog of the Bass–Quillen conjecture', Journal of the Institute of Mathematics of Jussieu 19 (2019), 1931–1946. DOI: 10.1017/S147474801900001X, arXiv:1608.00703, https://doi.org/10.1017/s147474801900001x
10. Mrinal Kanti Das, 'Remarks on Euler class groups and two conjectures', arXiv:1901.10686 (2019). https://arxiv.org/abs/1901.10686
11. Manoj K. Keshari, Swapnil A. Lokhande, 'Projective modules over overrings of polynomial rings and a question of Quillen', Journal of Pure and Applied Algebra 218 (2014), 1003–1011. DOI: 10.1016/j.jpaa.2013.10.015, https://doi.org/10.1016/j.jpaa.2013.10.015
12. Daniel Schäppi, 'Formal Laurent series rings and the Hermite ring conjecture', arXiv:2204.06303 (2022). https://arxiv.org/abs/2204.06303
13. Ning Guo, Fei Liu, 'The Bass–Quillen Conjecture for Torsors Over Valuation Rings', International Mathematics Research Notices (2025). DOI: 10.1093/imrn/rnaf142, arXiv:2503.10163, https://doi.org/10.1093/imrn/rnaf142
14. Anastasia Stavrova, 'On the generalized Bass–Quillen conjecture in dimension 2', arXiv:2512.18868 (2025). https://arxiv.org/abs/2512.18868
15. Tsit Yuen Lam, 'Serre's Problem on Projective Modules', Springer, Berlin Heidelberg (2006). DOI: 10.1007/978-3-540-34575-6, https://doi.org/10.1007/978-3-540-34575-6